Control strategies for transport networks under demand uncertainty

Abstract

In this article, we consider transport networks with uncertain demands. Network dynamics are given by linear hyperbolic partial differential equations and suitable coupling conditions, while demands are incorporated as solutions to stochastic differential equations. For the demand satisfaction, we solve a constrained optimal control problem. Controls in terms of network inputs are then calculated explicitly for different assumptions. Numerical simulations are performed to underline the theoretical results.

Appendix

We present the detailed calculation of the second moment for a Jacobi process with time-varying mean reversion level given by the SDE

$$ \begin{array}{@{}rcl@{}} dZ_{t} = \kappa (\theta(t) - Z_{t})dt + \sigma \sqrt{Z_{t}(1-Z_{t})}dW_{t}. \end{array} $$

(A.1)

Lemma A.1

Let $(\bar {\theta }_{n})_{n \in \mathbb {N}}$ be a sequence of step functions converging uniformly to a function 𝜃 ∈ C¹([t₀,T]). Additionally, let $\underset {n \rightarrow \infty }{\lim } t_{n} = T$. Then, the conditional second moment for the solution of (A.1) is given by

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[{Z_{T}^{2}} | Z_{t_{0}}=z_{0}] &=& {\int}_{t_{0}}^{T} (2\kappa \theta(s) + \sigma^{2})\left( z_{0}e^{-\kappa(s-t_{0})} + \kappa {\int}_{t_{0}}^{s} \theta(r)e^{-\kappa(s-r)}dr \right)e^{-(2\kappa + \sigma^{2})(T-s)}ds \\ &&~~~~+ {z_{0}^{2}}e^{-(2\kappa + \sigma^{2})(T-t_{0})}. \end{array} $$

Proof 1

For a constant mean reversion level, the conditional second moment is presented in equation (??). The idea of the proof is to use this expression to find a representation of the second moment for piecewise constant mean reversion levels and then use a uniform limit to show the result for continuously differentiable functions 𝜃. First, assume that for t₀ < t₁ < … < t_n on a bounded interval [t₀,t_n], 𝜃 is a step-function, i.e.,

$$ \begin{array}{@{}rcl@{}} \theta(t) = {\sum}_{i=0}^{n-1} \theta_{i} \mathbbm{1}_{[t_{i},t_{i+1})}(t) \end{array} $$

(A.2)

for $\theta _{i} \in \mathbb {R}$. For n = 1, we obtain the the conditional second moment as in (??). We use an induction to calculate the conditional second moment for an arbitrary $n+1 \in \mathbb {N}$ assuming that the conditional second moment is known for a step-function with n steps is given by

$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}[Z_{t_{n}}^{2} | Z_{t_{0}} = z_{0}] = {\sum}_{i=0}^{n-1}\Bigg[\frac{(2 \kappa \theta_{i} + \sigma^{2})\theta_{i}}{2 \kappa + \sigma^{2}} e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} + \frac{\kappa \theta_{i} (2 \kappa \theta_{i} + \sigma^{2})}{(2\kappa + \sigma^{2})(\kappa + \sigma^{2})} e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \Bigg]\\ &&~~~~ + {\sum}_{i=1}^{n-1} \Bigg[ \frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} \Bigg({\sum}_{j=1}^{i} (\theta_{j-1} - \theta_{j})e^{-\kappa(t_{i+1} - t_{j})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} -\theta_{i-1}e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \\ &&~~~~ - {\sum}_{j=1}^{i-1}(\theta_{j-1} - \theta_{j})e^{-\kappa(t_{i} - t_{j}) - (2\kappa + \sigma^{2})(t_{n} - t_{i})} \Bigg)\Bigg] \\ && ~~~~ + {\sum}_{i=1}^{n-1} \Bigg[\frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}}(z_{0}-\theta_{0})\left( e^{-\kappa(t_{i+1} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} -e^{-\kappa(t_{i} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i})}\right)\Bigg] \\ &&~~~~+\frac{2\kappa \theta_{0} + \sigma^{2}}{\kappa + \sigma^{2}} \left( (z_{0} - \theta_{0})e^{-\kappa(t_{1} - t_{0}) - (2\kappa + \sigma^{2})(t_{n}-t_{1})} - z_{0} e^{-(2\kappa + \sigma^{2})(t_{n}-t_{0})}\right) + {z_{0}^{2}}e^{-(2\kappa + \sigma^{2})(t_{n} - t_{0})}. \end{array} $$

(A.3)

Then the induction step to n + 1 reads

$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left[Z_{t_{n+1}}^{2} | Z_{t_{0}}= z_{0} \right] = \mathbb{E}\left[Z_{t_{n+1}}^{2} - Z_{t_{n}}^{2} + Z_{t_{n}}^{2} | Z_{t_{0}}= z_{0} \right] = \mathbb{E}\left[\mathbb{E}\left[Z_{t_{n+1}}^{2} | Z_{t_{n}} = Z_{n}\right] ~| Z_{t_{0}}= z_{0} \right] \\ &&= \mathbb{E} \Bigg[\frac{(2\kappa \theta_{n} + \sigma^{2})\theta_{n}}{2\kappa + \sigma^{2}} + \frac{2\kappa \theta_{n} + \sigma^{2}}{\kappa + \sigma^{2}}(Z_{n} - \theta_{n}) e^{-\kappa(t_{n+1}-t_{n})} \\ &&~~~~+ \left( {Z_{n}^{2}} - \frac{2\kappa\theta_{n} + \sigma^{2}}{\kappa + \sigma^{2}}Z_{n} + \frac{\kappa \theta_{n} (2\kappa \theta_{n} + \sigma^{2})}{(2\kappa + \sigma^{2})(\kappa + \sigma^{2})} \right)e^{-(2\kappa + \sigma^{2})(t_{n+1}-t_{n})} \Big| Z_{t_{0}}=z_{0} \Bigg] \\ &&= \frac{(2\kappa \theta_{n} + \sigma^{2})\theta_{n}}{2\kappa + \sigma^{2}}+ \frac{2\kappa \theta_{n} + \sigma^{2}}{\kappa + \sigma^{2}} \bigg({\sum}_{j=1}^{n} (\theta_{j-1} - \theta_{j})e^{-\kappa(t_{n+1} - t_{j})} \bigg) \\ &&~~~~- \frac{2\kappa \theta_{n} + \sigma^{2}}{\kappa + \sigma^{2}}\bigg(\theta_{n-1}e^{-(2\kappa + \sigma^{2})(t_{n+1} - t_{n})}+ {\sum}_{j=1}^{n-1} (\theta_{j-1} - \theta_{j})e^{-\kappa(t_{n}-t_{j}) - (2\kappa + \sigma^{2})(t_{n+1} - t_{n})} \bigg) \\ &&~~~~+ \frac{\kappa\theta_{n}(2\kappa \theta_{n} + \sigma^{2})}{(2\kappa + \sigma)(\kappa + \sigma^{2})} e^{-(2\kappa + \sigma^{2})(t_{n+1} - t_{n})} + \frac{2\kappa \theta_{n} + \sigma^{2}}{\kappa + \sigma^{2}}\left( z_{0}-\theta_{0}\right)\left( e^{-\kappa(t_{n+1} - t_{0})} - e^{-\kappa(t_{n} - t_{0})}\right) \\ &&~~~~ + \Bigg[{\sum}_{i=0}^{n-1}\Bigg[\frac{(2 \kappa \theta_{i} + \sigma^{2})\theta_{i}}{2 \kappa + \sigma^{2}} e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} + \frac{\kappa \theta_{i} (2 \kappa \theta_{i} + \sigma^{2})}{(2\kappa + \sigma^{2})(\kappa + \sigma^{2})} e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \Bigg]\\ &&~~~~ + {\sum}_{i=1}^{n-1} \Bigg[ \frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} \Big({\sum}_{j=1}^{i} (\theta_{j-1} - \theta_{j})e^{-\kappa(t_{i+1} - t_{j})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} \Big) \\ &&~~~~- \frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} \Big(\theta_{i-1}e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})}+ {\sum}_{j=1}^{i-1}(\theta_{j-1} - \theta_{j})e^{-\kappa(t_{i} - t_{j}) - (2\kappa + \sigma^{2})(t_{n} - t_{i})} \Big) \Bigg] \\ && ~~~~ + {\sum}_{i=1}^{n-1} \Bigg[\frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} \Bigg((z_{0}-\theta_{0})\left( e^{-\kappa(t_{i+1} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} -e^{-\kappa(t_{i} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i})}\right) \Bigg) \Bigg]\\ &&~~~~ +\frac{2\kappa \theta_{0} + \sigma^{2}}{\kappa + \sigma^{2}} \left( (z_{0} - \theta_{0})e^{-\kappa(t_{1} - t_{0}) - (2\kappa + \sigma^{2})(t_{n}-t_{1})} -z_{0} e^{-(2\kappa + \sigma^{2})(t_{n}-t_{0})}\right) \\ &&~~~~+ {z_{0}^{2}}e^{-(2\kappa + \sigma^{2})(t_{n} - t_{0})} \Bigg]e^{-(2\kappa + \sigma^{2})(t_{n+1} - t_{n})}. \end{array} $$

Summarizing, we obtain the proposed equation (A.3). As a last step, we calculate the limit for $n \rightarrow \infty$.

$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}[{Z_{T}^{2}} | Z_{t_{0}} = z_{0}] \\&&= \underset{n \rightarrow \infty}{\lim} ~ {\sum}_{i=0}^{n-1}\Bigg[\frac{(2 \kappa \theta_{i} + \sigma^{2})\theta_{i}}{2 \kappa + \sigma^{2}} e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} + \frac{\kappa \theta_{i} (2 \kappa \theta_{i} + \sigma^{2})}{(2\kappa + \sigma^{2})(\kappa + \sigma^{2})} e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \Bigg]\\ &&~~~~ + {\sum}_{i=1}^{n-1} \Bigg[ \frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} \Bigg({\sum}_{j=1}^{i} (\theta_{j-1} - \theta_{j})e^{-\kappa(t_{i+1} - t_{j})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} -\theta_{i-1}e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \\ &&~~~~ - {\sum}_{j=1}^{i-1}(\theta_{j-1} - \theta_{j})e^{-\kappa(t_{i} - t_{j}) - (2\kappa + \sigma^{2})(t_{n} - t_{i})} \Bigg)\Bigg] \\ && ~~~~ + {\sum}_{i=1}^{n-1} \Bigg[\frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}}(z_{0}-\theta_{0})\left( e^{-\kappa(t_{i+1} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} -e^{-\kappa(t_{i} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i})}\right)\Bigg] \\ &&~~~~+\frac{2\kappa \theta_{0} + \sigma^{2}}{\kappa + \sigma^{2}} \left( (z_{0} - \theta_{0})e^{-\kappa(t_{1} - t_{0}) - (2\kappa + \sigma^{2})(t_{n}-t_{1})} -z_{0} e^{-(2\kappa + \sigma^{2})(t_{n}-t_{0})}\right) \\ &&~~~~+ {z_{0}^{2}}e^{-(2\kappa + \sigma^{2})(t_{n} - t_{0})}\\ &&= \frac{(2 \kappa \theta(t_{0}) + \sigma^{2})\theta(t_{0})}{2 \kappa + \sigma^{2}} \left( 1+ \frac{\kappa}{\kappa + \sigma^{2}} \right) e^{-(2\kappa + \sigma^{2})(T - t_{0})} - \frac{2\kappa \theta(t_{0}) + \sigma^{2}}{\kappa + \sigma^{2}} \theta(t_{0})e^{-(2\kappa + \sigma^{2})(T - t_{0})} \\ &&~~~~+ \underset{n \rightarrow \infty}{\lim} \Bigg[ {\sum}_{i=1}^{n-1}\Bigg[\frac{2 \kappa \theta_{i} + \sigma^{2}}{(2 \kappa + \sigma^{2})(\kappa + \sigma^{2})} \bigg((\kappa + \sigma^{2}) \Big(\theta_{i} e^{-(2 \kappa + \sigma^{2})(t_{n} - t_{i+1})} - \theta_{i-1} e^{-(2 \kappa + \sigma^{2})(t_{n} - t_{i})}\Big) \\ &&~~~~ + \kappa \left( \theta_{i} - \theta_{i-1} \right)e^{-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \bigg) \Bigg]+ {\sum}_{i=1}^{n-1}\frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} (\theta_{i-1} - \theta_{i})e^{- (2\kappa + \sigma^{2})(t_{n} - t_{i+1})} \\ &&~~~~ + {\sum}_{i=1}^{n-1} \frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}} {\sum}_{j=1}^{i-1}(\theta_{j-1} - \theta_{j})\left( e^{-\kappa(t_{i+1} - t_{j}) - (2\kappa + \sigma^{2})(t_{n} - t_{i+1})} - e^{-\kappa(t_{i} - t_{j}) - (2\kappa + \sigma^{2})(t_{n} - t_{i})} \right) \\ &&~~~~ + {\sum}_{i=1}^{n-1} \Bigg[\frac{2\kappa \theta_{i} + \sigma^{2}}{\kappa + \sigma^{2}}(z_{0}-\theta_{0})\left( e^{-\kappa(t_{i+1} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i+1})} - e^{-\kappa(t_{i} - t_{0})-(2\kappa + \sigma^{2})(t_{n} - t_{i})} \right)\Bigg]\Bigg] \\ &&~~~~+ {z_{0}^{2}}e^{-(2\kappa + \sigma^{2})(T - t_{0})}\\ &&= \frac{(2 \kappa \theta(t_{0}) + \sigma^{2})\theta(t_{0})}{\kappa + \sigma^{2}} e^{-(2\kappa + \sigma^{2})(T - t_{0})}- \frac{2\kappa \theta(t_{0}) + \sigma^{2}}{\kappa + \sigma^{2}}\theta(t_{0})e^{-(2\kappa + \sigma^{2})(T - t_{0})} \\ &&~~~~+ {\int}_{t_{0}}^{T}\frac{2 \kappa \theta(s) + \sigma^{2}}{(2 \kappa + \sigma^{2})(\kappa + \sigma^{2})} \left( (2\kappa + \sigma^{2}) \left( \theta^{\prime}(s) + (2\kappa + \sigma^{2})\theta(s)\right) \right)e^{-(2\kappa + \sigma^{2})(T-s)}ds \\ &&~~~~ + {\int}_{t_{0}}^{T} \frac{2\kappa \theta(s) + \sigma^{2}}{\kappa + \sigma^{2}} \left( -\theta^{\prime}(s) + {\int}_{t_{0}}^{s} -\theta^{\prime}(r)(\kappa + \sigma^{2})e^{-\kappa(s-r)}dr \right)e^{- (2\kappa + \sigma^{2})(T - s)}ds\\ &&~~~~+ {\int}_{t_{0}}^{T} (2\kappa \theta(s) + \sigma^{2})(z_{0}-\theta(t_{0})) e^{-\kappa(s - t_{0})-(2\kappa + \sigma^{2})(T - s)} ds+ {z_{0}^{2}}e^{-(2\kappa + \sigma^{2})(T - t_{0})}\\ &&= {\int}_{t_{0}}^{T} (2\kappa \theta(s) + \sigma^{2})\Big(\theta(s) - {\int}_{t_{0}}^{s} \theta^{\prime}(r)e^{-\kappa(s-r)}dr +(z_{0} - \theta(t_{0}))e^{-\kappa(s-t_{0})} \Big)e^{-(2\kappa + \sigma^{2})(T-s)} ds\\ &&~~~~+ {z_{0}^{2}} e^{-(2\kappa + \sigma^{2})(T - t_{0})}\\ &&= {\int}_{t_{0}}^{T} (2\kappa \theta(s) + \sigma^{2})\left( z_{0}e^{-\kappa(s-t_{0})} + \kappa {\int}_{t_{0}}^{s} \theta(r)e^{-\kappa(s-r)}dr\right) e^{-(2\kappa + \sigma^{2})(T-s)}ds \\ &&~~~~+{z_{0}^{2}} e^{-(2\kappa + \sigma^{2})(T-t_{0})} \end{array} $$

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